scholarly journals Going up and lying over in congruence-modular algebras

2019 ◽  
Vol 69 (2) ◽  
pp. 275-296
Author(s):  
George Georgescu ◽  
Claudia Mureşan
Keyword(s):  
General Setting ◽  
Ring Theory ◽  
Direct Products ◽  
Mv Algebras

Abstract In this paper, we extend properties Going Up and Lying Over from ring theory to the general setting of congruence-modular equational classes, using the notion of prime congruence defined through the commutator. We show how these two properties relate to each other, prove that they are preserved by finite direct products and quotients and provide algebraic and topological characterizations for them. We also point out many kinds of varieties in which these properties always hold, generalizing the results of Belluce on MV-algebras and Rasouli and Davvaz on BL-algebras.

Lifting components in clean abelian ℓ-groups

2018 ◽  
Vol 68 (2) ◽  
pp. 299-310
Author(s):  
Karim Boulabiar ◽  
Samir Smiti
Keyword(s):  
Vector Lattice ◽  
General Setting ◽  
Ring Theory ◽  
Natural Analogue ◽  
Order Unit ◽  
Step Functions ◽  
Strong Order

Abstract Let G be an abelian ℓ-group with a strong order unit u > 0. We call G u-clean after Hager, Kimber, and McGovern if every element of G can be written as a sum of a strong order unit of G and a u-component of G. We prove that G is u-clean if and only if u-components of G can be lifted modulo any ℓ-ideal of G. Moreover, we introduce a notion of u-suitable ℓ-groups (as a natural analogue of the corresponding notion in Ring Theory) and we prove that the ℓ-group G is u-clean when and only when it is u-suitable. Also, we show that if E is a vector lattice, then E is u-clean if and only if the space of all u-step functions of E is u-uniformly dense in E. As applications, we will generalize a result by Banaschewski on maximal ℓ-ideals of an archimedean bounded f-algebras to the non-archimedean case. We also extend a result by Miers on polynomially ideal C(X)-type algebras to the more general setting of bounded f-algebras.

scholarly journals Subalgebras, direct products and associated lattices of MV-algebras

1992 ◽  
Vol 34 (3) ◽  
pp. 301-307 ◽  
Author(s):  
L. P. Belluce ◽  
A. Di Nola ◽  
A. Lettieri
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Propositional Logic ◽  
Abelian Groups ◽  
Completeness Theorem ◽  
Direct Products ◽  
Algebraic Proof ◽  
Mv Algebras

MV-algebras were introduced by C. C. Chang [3] in 1958 in order to provide an algebraic proof for the completeness theorem of the Lukasiewicz infinite valued propositional logic. In recent years the scope of applications of MV-algebras has been extended to lattice-ordered abelian groups, AF C*-algebras [10] and fuzzy set theory [1].

scholarly journals On (finite) distributive lattices with antitone involutions

10.29007/81mc ◽  
2018 ◽  
Author(s):  
Jan Kühr ◽  
Michal Botur
Keyword(s):  
Distributive Lattices ◽  
Direct Products ◽  
Natural Conditions ◽  
Finite Distributive Lattices ◽  
Mv Algebras

Finite distributive lattices with antitone involutions (= basic algebras) are studied; it is proved that their underlying lattices are isomorphic to direct products of finite chains, and hence finite distributive basic algebras can be constructed by “perturbing” finite MV-algebras, and moreover, under certain natural conditions, they even coincide with finite MV-algebras.

FPF Ring Theory

1984 ◽  
Author(s):  
Carl Faith ◽  
Stanley Page
Keyword(s):  
Ring Theory

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

Semi-simple and complete MV-algebras

1992 ◽  
Vol 29 (1) ◽  
pp. 1-9 ◽  
Author(s):  
L. P. Belluce
Keyword(s):  
Mv Algebras
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